Numerical Methods in Soil Mechanics 11.PDF

Numerical Methods in Soil Mechanics 11.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.

Anderson, Loren Runar et al "ENCASED FLEXIBLE PIPES"

Structural Mechanics of Buried Pipes

Boca Raton: CRC Press LLC,2000

Figure 11-1 Inserted encased pipe (liner) showing how pressure develops between the liner and the encasement

Figure 11-2 Basic deformation of the encased, flexible liner showing how the ring is blown against the encasement such that a gap opens and a blister forms in the liner

CHAPTER 11 ENCASED FLEXIBLE PIPES

Encased pipes include the following:

a) Flexible pipe with concrete cast about it,

b) Pipe inserted into another pipe or a tunnel,

c) Liner for deteriorated pipe,

d) Liner in an encasement in which grout fills the

annular space between pipe and encasement

All have a common performance limit — ring

inversion due to external pressure A special case

is a pipe that floats in fluid (water or grout) in the

encasement For equations of analysis, see

Appendix A Two basic analyses are: liner and

double wall

LINER ANALYSIS

The inside pipe is the liner It is so flexible

compared to the encasement that it can be analyzed

as a flexible ring in a rigid encasement

Performance limit is inversion of the liner Collapse

may be time dependent based on "creeping" ring

deformation under persistent pressure Collapse is

sudden inversion Resistance to inversion is ring

stability — a function of yield strength, σf, ring

stiffness, EI/r3, and ring deflection, d It is assumed

that pressure persists against the ring

Internal Pressure Failure of Encased Rings

Internal persistent pressure at fracture is a special

case of instability Once the ring starts to yield, the

diameter increases and the rupturing force in the

wall increases If the liner is designed to take the

internal pressure, there is no failure If the rigid

encasement is designed to take the internal pressure,

there is no failure The liner is an innertube If both

liner and encasement expand, each shares in

resisting the internal pressure For analysis, the

relative resistances of each to expansion by internal

pressure must be known It is conservative to

design both encasement and liner so that each can

take the full internal pressure

External Pressure at Inversion of Encased Rings The performance limit is inversion of the liner due to

external pressure Because encasements usually

leak, if the water table is above the encasement, water pressure builds up between encasement and liner See Figure 11-1 Even if the liner is bonded to the encasement, water pressure peels the liner away from the encasement and deforms the liner as shown in Figure 11-2 A gap forms where the liner

is unencased — over an arc no greater than 180o

The unencased section is a blister — between

circular and approximately elliptical Pressure P at inversion is greater than it would be if the liner were completely unencased But part of the liner is unencased

For a completely unencased liner, the classical

analysis, from Chapter 10, is,

Pc rr3/EI = 3 (11.1) UNENCASED CIRCULAR RING COLLAPSE

Pc r applies to a circular ring

For a deformed ring, elliptical analysis, from Chapter

10, is:

P2 - [σf /m+(1+6mdo)Pc r]P + σf P c r /m = 0

(10.3)

Notation is listed in the paragraph, "Arc Angle 2α."

Figure 11-3 shows plots of Equation 10.3 for a plain steel pipe and a plain PVC plastic pipe Because the PVC dimension ratio is DR = OD/t, it follows that 2m = DR-1 for ring flexibility of the PVC pipe

From Figure 11-3 it is evident that the effect of ring

deflection on P is significant for an unencased ring However, for an unencased blister in a liner, the

difference between circle and ellipse is not significant One exception is a slip liner smaller in diameter than the encasement In the following, the unencased section of liner is circular

Steel Pipe

σf = 42 ksi

E = 30(103) ksi

PVC Pipe

σf = 4 ksi

E = 400 ksi

Figure 11-3 Examples of external pressure at collapse of unencased pipes with elliptical cross sections.

Figure 11-4 is a summary of inversion mechanisms

f or encased rings If the gap is small (blister does

not form) ring compression applies If a blister

forms over the gap, it acts either as an arch o r a

beam Analysis of beam failure is classical An

unconfined ar ch could potentially invert If the gap

is small, arch instability is analyzed for a circular arc

If the gap is large, the arch is elliptical A simple,

but conservative analysis is circular analysis using

the maximum radius of curvature of the ellipse

Arc Angle 2α

An unknown is the arc angle, 2α Figure 11-5

shows the collapse of a flexible, circular, hinged

arch From Timoshenko (1956),

Pr3/EI = (π/α) 2 - 1 (11.2)

Notation:

P = pressure on the blister at inversion,

D = mean circular diameter of the liner,

r = mean radius of the circular liner,

t = wall thickness,

m = r/t = ring flexibility,

E = modulus of elasticity,

σf = yield strength of the liner,

∆ = decrease in minimum diameter,

d = ∆/D = ring deflection,

T = circumferential thrust in liner wall,

M = moment due to ring deformation,

α = half arc angle (Figure 11-5),

β = blister angle (gap angle),

σ = maximum circumferential stress,

h = height of water table above the liner,

H = height of soil cover over the pipe,

I = moment of inertia of wall cross section

Equation 11.2 can be applied to an encased flexible

ring by selecting a portion of the ring that is

equivalent to the circular hinged arch of Figure 11-5

Figure 11-6 shows the typical inversion of a flexible

liner The blister can be seen developing in the

blister arc, β At the ends of the arc, the moment is

maximum, and plastic hinges can be seen

developing These become the gunwales of an

inverted "boat hull" that rises up into the pipe A third plastic hinge forms at the keel Points of zero moment are circles Points of maximum moment are triangles

Points B (circles) are points of tangency to the encasement — and so moment is zero

Points C (circles) are points of counterflexure — and so are hinge points — zero moment

Points ∆ (triangles) are plastic hinges — points of maximum moment that isolate the collapse mechanism in arc, β

Points B, C, and ∆ are about equally spaced Therefore, β = 2α, which is equivalent to the arc angle of Figure 11-5, and can be analyzed by Equation 11.2, from which critical P is,

P = E[(π/α) 2-1]/12m3 (11.3) Angle α is unknown From tests, the arc angle for plastic liners is roughly, α = 30o to 45o Equation 11.3 neglects decrease in circumference due to ring compression In order to find α, worst case assumptions are as follows:

Assumptions:

1 There is no bond, interlocking, or frictional resistance between liner and the encasement

2 There is no pressure inside the liner

3 The liner is subjected to external pressure P Leaks in the encasement allow pressure on the liner due to groundwater table External pressure includes any vacuum that may occur inside the liner

4 The liner is flexible Initially, it fits snugly against the encasement But it may shrink, leaving

an annular space between liner and encasement The liner is snug (but not press-fit) in the encasement

5 The liner may be plastic which can creep under persistent pressure over a period of time

6 The cross section of the blister is circular Third-dimensional (longitudinal) resistance to the formation of a blister is neglected

Figure 11-4 Inversion mechanisms for a blister in a liner

Figure 11-5 Collapse of a hinged circular arch subjected to uniform radial pressure, P.

Figure 11-6 Typical inversion of a flexible liner at critical pressure P β = blister angle

When subjected to external pressure, the liner

shrinks Bond breaks down between liner and

encasement External pressure, P, distributes itself

around the entire surface of the liner At one

location a gap opens between liner and encasement,

and a "blister" forms in the liner The blister

develops wherever the radius is maximum, or at the

bottom of the liner where external hydrostatic

pressure is greatest Failure is inversion of the

blister From inside the pipe the inversion looks like

a boat hull with the keel longitudinal

Ring compression thrust, Pr, is constant all around

the ring Assuming P is constant, ring compression

thrust is Pry, where ry is the greatest radius of

curvature

Analysis is prediction of minimum pressure P at

inversion See Figure 11-4 The rationale is to

calculate the decrease in perimeter of the liner, and

to find ry as a function of the arc angle It is then

possible to find P, both when ring compression stress

is at yield, and when the blister inverts The lesser

of these two P's is critical

Notation:

r = constrained mean radius of the liner ring,

ry = maximum radius of the blister,

OD = outside diameter of the unpressurized liner,

t = thickness of the liner wall,

A = area of the liner per unit length of pipe,

A = t for a plain liner,

δ = decrease in circumference of the liner ring,

DR = dimension ratio of the liner = OD/t,

α = half arc angle based on casing radius r,

β = half arc angle based on blister radius, ry,

E = modulus of elasticity of the liner,

σ = ring compression stress in the liner,

σf = yield stress, (at 50 years of persistent

pressure?)

Inversion is by one of three mechanisms shown in

Figure 11-4:

a) Wall crushing if ring compression exceeds yield,

b) Arch inversion,

c) Beam failure if the blister is flat

The ends of the beam are the plastic hinges at the edges of the blister The moment capacity of plastic hinges is, Mp = 3Me /2, where Me is the elastic moment at yield stress; i.e., Me = σf I/c

Wall Crushing Analysis

The liner buckles when ring compression stress equals yield; i.e., σ = σf, where σ = Pr/A For plain liners,

P = σf /(ry /t) (11.4)

where ry is the maximum radius (which occurs at the blister) where the maximum stress, is Pry /t See

Figure 11-7 (top) But Pr/t is constant all around the ring Therefore, the decrease in perimeter of the liner due to ring compression strain is,

The geometrical decrease in perimeter of the liner is:

δ = 2rα - 2ryβ (11.6)

The blister width is AB = 2rsinα = 2rysinβ, from which ry = rsinα/sinβ Equating the decreases in perimeter from Equations 11.5 and 11.6,

β/sinβ = α/sinα - πσf /Esinα (11.7)

which can be solved by iteration for β in terms of α

Of interest is angle α at flat blister (straight beam at

β = 0), for which β/sinβ = 1

From Figure 11-7, blister width AB = 2rsinα = 2rysinβ Substituting the resulting ry = rsinα /sinβ

into Equation 11.4, critical pressure is,

P = σfsinβ/(r/t)sinα (11.8)

Figure 11-7 (bottom) shows pressure P = f(α ) The

inversion sequence starts with ring compression

yielding (wall crushing), P = σf /(r/t), at α = β A

blister forms at yield stress, σf With its greater

r a d i u s , ry, the blister rises and the blister angle

decreases But yielding is not inversion Yielding

progresses down the wall crushing curve from right

to left — shown by arrows Based only on wall

crushing, the blister inverts at the least value of α

In the example of Figure 11-7, α = 33o But before

the blister inverts by ring compression, it might invert

by arch inversion or by beam failure In Figure 11-7,

the arch inverts at α = 34o

Arch Inversion Analysis

From Equation 11.3, for a hinged circular arch,

critical P is a function of arch angle α From Figure

11-6, α is a third of the gap angle, B-B, which

cannot exceed 180o The plastic pipe example of

Figure 11-7 shows arch inversion as a heavy solid

line P increases as α decreases But α decreases

only by ring compression strain (down the wall

crushing curve) So α is critical where wall

crushing intersects the arch inversion curve

Inversion occurs at α = 34o, and P = 57 psi For

very conservative analysis, set α = 60o (one-third of

the 180o upper limit of gap angle)

Beam Failure Analysis

For low strength materials, inversion may be beam

failure At inversion, the blister cross section is a

fixed-ended beam With the length known from

radius r and angle α, pressure P can be found from

the equation of stress,

σ = Me c/I

where M = PL2/12 for a fixed-end beam (11.9)

But plastic hinges form at Mp = 3Me /2, Substituting,

9σf = 2Pr2sin2α /(I/c) (11.10) For plain liners (no ribs), I/c = t2/6, and (r/t) = m Substituting into Equation 11.10,

3σf = 4P(r/t)2sin2α, from which, at inversion

P = 3σf /4m2sin2α (11.11) Example

A hypothetical plastic liner has the following properties:

DR = 51,

m = r/t = 25,

E = 400 ksi = virtual modulus of elasticity over

50 years of persistent pressure,

σf = 4 ksi = yield strength at 50 years of

pressure

Find: Persistent pressure P that would cause inversion at 50 years of service

From Equation 11.7, due to ring compression strain,

β/sinβ = α /sinα - πσf /Esinα For trial values of α, corresponding values of β are found by iteration Of course, ring compression stress approaches infinity when the blister is flat; i.e., when β/sinβ = 1 Equation 11.7 becomes, α /sinα - πσf /Esinα = 1 Solving, at inversion, α = 33o But inversion occurs

by arch inversion or beam failure before α shrinks to

33o From Equation 11.8,

P = σf sinβ/msinα (11.8)

Figure 11-7 is a plot of Equation 11.8, showing long-term critical P as a function of α Quick wall

c rushing would occur at P = 160 psi as show n Over the long term, the plastic ring creeps, α is reduced, and, therefore, inversion pressure, P, is reduced The amount of creep is found from long-term tests For most plastics, failure is arch inversion From Figure 11-7 inversion occurs at α =

34o Critical pressure is, P = 57 psi

Figure 11-7 Example of graphs of critical pressure, P, as a function of the half blister angle, α, for three collapse mechanisms in a typical plastic pipe Yielding (wall crushing) may be time dependent because of plastic creep

Only in rare cases, such as low strength or

undersized liners, will inversion be beam failure As

an exercise only, if a blister in the liner could flatten

into a beam, at α = 33o from Figure 11-7,

from Equation 11.11, P = 16.2 psi at 50 years of

persistent pressure

The above procedure provides an approximate

inversion analysis for liners It is a limit analysis

Critical pressure at inversion is greater than the

c alculated pressure because of longitudinal

resistance, bond, etc Compared to tests, analysis is

conservative

The analysis allows for modifications and

innovations For example, when shrinkage of the

liner is due to conditions other than external

pressure, such differences can be included in the

analysis One modification is the virtual modulus of

elasticity E (not actual modulus — but, mistakenly,

referred to as long-term modulus) that allows for

creep of the liner over a long period of time This

creep causes a decrease in perimeter of the liner

under constant pressure

If the liner is initially out-of-round, a gap will form

where the radius is maximum Using maximum

radius instead of the circular radius, analysis of the

liner can proceed Radius of curvature can be

calculated from a measured offset from the

midlength of a cord placed across the curved section

of liner

The procedure can be programmed for computer

It can also be presented as tables and graphs which,

for most overworked engineers, may be the best

presentation

Plastic pipes are often used as liners for

rehabilitating damaged pipelines, usually in a full

contact fit in the damaged encasement

Example

A folded PVC pipe is inserted, heated and inflated to

become a liner in an 8 ID encasement Find critical

pressure P

P = external pressure at inversion,

E = 420 ksi = modulus of elasticity,

σf = 3 ksi = yield strength,

DR = 35 = standard dimension ratio,

m = 17 = r/t = (DR-1)/2,

ν = 0.38 = Poisson ratio

In the short term, critical pressure is P = σf /m = 176 psi From eight tests, the critical pressure was 172 psi with a standard deviation of 38 psi

If unencased, the maximum external collapse pressure is only

P = E/4 m3(1-ν 2) = 25 psi Clearly the observed collapse pressure is much closer to ring compression theory than to the hydrostatic collapse theory It is noteworthy that the test liners failed by bulging inward throughout a blister angle less than about β = 90o in Figure 11-6 For a blister angle of 60o, α = 30o From Equation 11.3, arch inversion occurs at P = 107 psi But this

is based on E = 420 ksi Over the long term, if virtual E were only two-thirds as great, critical pressure would be only 71 psi

COMPARISON OF ANALYTICAL METHODS

The following is a rough comparison of some methods proposed for analyzing the constrained flexible pipes subjected to external pressure Soil contributes significant constraint to the buried flexible ring subjected to uniform external pressure Two equations used in service for design of rings that are circular or nearly circular (encased), are as follows

a) One form of the AWWA C950 formula in AWWA-M11 (1989) is,

P2 = 0.593RwEsEI/0.149r3

where

Rw = buoyancy factor = 1-0.33(h/H),

Es = soil stiffness = secant modulus